Producer willingness to supply biomass. The conversion of biomass to energy is critical for producing transportation fuels. How willing are producers to supply biomass products such as cereal straw, corn stover, and surplus hay? Economists surveyed producers in both mid-Missouri and southern Illinois (Biomass and Energy, Vol. 36, 2012). Independent samples of 431 Missouri producers and 508 Illinois producers participated in the survey. Each producer was asked to give the maximum proportion of hay produced that they would be willing to sell to the biomass market. Summary statistics for the two groups of producers are listed in the table. Does the mean amount of surplus that hay producers are willing to sell to the biomass market differ for the two areas, Missouri and Illinois? Use a = .05 to make the comparison.

Agriculture and Industry in the Missouri economy is a set of human and social activities and institutions related to the production, distribution, exchange, and consumption of agriculture and industry goods and services.

In this exercise, we will determine if the mean amount of surplus that hay produced is willing to sell to the biomass market differs for the two areas (Missouri and Illinois) by using the $z$-test for two population means.

Since the sample size of the Missouri producers and Illinois Producers are greater than 30, then $z$-test for two population means must be used in this hypothesis testing.

Assume that Sample 1 consists of the Missouri Producers and Sample 2 consists of the Illinois Producers. Then we can form the null and alternative hypotheses and conduct the hypothesis testing of this study.

This study's null as well as alternative hypotheses are as follows:

${\mathrm{H}}_0$: There is no significant difference in the mean amount of surplus that hay produced is willing to sell to the biomass market between the two areas $({\mathrm{\mu }}_1-{\mathrm{\mu }}_2=0)$.

${\mathrm{H}}_{\mathrm{a}}$: There is a significant difference in the mean amount of surplus that hay produced is willing to sell to the biomass market between the two areas$({\mathrm{\mu }}_1-{\mathrm{\mu }}_2¹0)$.

The test statistic ($\mathrm{z}$-value) is calculated by using this formula:

$z=\frac{({\overline{x}}_1-{\overline{x}}_2)-D_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$

Given the below:

$\mathrm{n}\_\left\{1\right\}=431$

${\overline{\mathrm{x}}}_1=21.5$

$s_1^2=33.4^2=1,115.56$

By looking at the normal curve areas (table II), the $p$-value $z=0.31$ is $0.1217.$The $p$-value of this two-tailed test $(z\ne 0.31)$is calculated below:

$p$-value ${}_{z¹0.31}=(0.5-0.1217)\times 2$

role="math" localid="1652699270623" $\begin{array}{l}=0.3783\times 2\\ =0.7566\end{array}$

Decision

Since the $p$-value$z¹0.31(0.7566)$ is greater than the level of significance,$(\alpha =.05),$ then the evidence to reject the null hypothesis is insufficient.

Therefore, there is no significant difference in the mean amount of surplus that hay produced is willing to sell to the biomass market between the two areas (Missouri and Illinois).